Telling the batter the count

Question
Last night my partner and I did a junior league double header. In the second game he had the plate.  Runner on first, one out. Count of 1 ball and 1 strike on the batter. The batter swung and missed the next pitch, it hit the ground and went behind the catcher.  The coach in the dugout shouted "run, run"...he was shouting to the baserunner, who just stayed on the bag.  The batter took off for first. Now forget all the "runner on first, less than two outs he can't run on a dropped third strike" because it was only strike 2.  My partner shouted to the batter "1 ball 2 strikes".  He went to first anyway.  The runner on first was confused, and got no help from the 1st base coach, he started jogging to second.  An alert catcher threw the ball to 2nd base and they tagged the runner.  I called him out.  The coach of course wanted to know what was going on.  I told him the runner was thrown out trying to steal second and the batter has a count of 1 ball and 2 strikes.  
Now for my question: What responsibility does an umpire have to stop stupid actions by the batter.  I believe my partner did all he should have done, letting the batter know the count.  I tell the umpires in my league to say "ball 4" not to point to first...the batters should be coached properly.  

Answer
I think you and your partner handled the situation in a text book manner.  Every level I have ever called, little league up through Conference USA, has directed the umpires to give the count after the 3rd and 5th pitches and whenever there could be some confusion (i.e. an scoreboard keeper makes what should be 2 and 2 read 3 and 1).  It is not our responsibility to coach the kids, simply to see that a fair game is played by the appropriate rules.  When your partner audibly gave the count, he did more than he "had" to right there.  So, all in all, everything sounds like you did your job.  

P.S. You're absolutely right in telling your umpires not to point to the base on ball four.  Directing players what to do in a situation (go to first) is coaching and we're there to umpire, not coach.